Hajer Bahouri

High frequency approximation of solutions to critical nonlinear wave equations

This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □u + |u|4 = 0 in [inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=01i /], up to remainder terms small in energy norm and in every Strichartz norm. The proof relies on scattering theory for (1) and on a structure theorem for bounded energy sequences of solutions to the linear wave equation. In particular, we infer the existence of an a priori estimate of Strichartz norms of solutions to (1) in terms of their energy.

Decay estimates for the critical semilinear wave equation

Abstract In this paper we prove that finite energy solutions (with added regularity) to the critical wave equation □u + u5 = 0 on R 3 decay to zero in time. The proof is based on a global space-time estimate and dilation identity.

Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg

In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.

Littlewood–Paley Theory

In Chapter 2 we give a detailed presentation on Littlewood-Paley decomposition and define homogeneous and nonhomogeneous Besov spaces. We should emphasize that we have replaced the usual definition of homogeneous spaces (which are quotient distribution spaces modulo polynomials) by something better adapted to the study of partial differential equations (indeed, dealing with distributions modulo polynomials is not appropriate in this context). We also establish technical results (commutator estimates and functional inequalities, in particular) which will be used in the following chapters.

Concentration effects in critical nonlinear wave equation and scattering theory

This paper is devoted to generalized geometrical optics for the following nonlinear wave equation with critical exponent, n n

Transport and Transport-Diffusion Equations

square u + left| u right|^4 u = 0

Strichartz Estimates and Applications to Semilinear Dispersive Equations

n n(1) n nwhere (square = partial _t^2 - Delta _x ,t in {text{R,}},x in {text{R}}^{text{3}} ). The global well-posedness of equation (1) in the energy space was proved rather recently by Shatah-Struwe [16]. Let us recall precisely this result: given (phi in dot H^1 left( {{text{R}}^{text{3}} } right),psi in L^2 left( {{text{R}}^{text{3}} } right)) there exists a unique solution u to (1) satisfying (uleft| {_{t = 0} = phi ,,partial _t u} right.left| {_{t = 0} = psi } right.)and (u in L_{{text{loc}}}^5 left( {{text{R}},L^{10} left( {{text{R}}^{text{3}} } right)} right)). Observe that this latter property means exactly that the nonlinear term (left| u right|^4 u) in (1) belongs to (L_{{text{loc}}}^1 left( {{text{R,}},L^2 left( {{text{R}}^{text{3}} } right)} right)), which allows us to consider it as a source term in the energy method. In particular, this implies (left( {u,partial _t u} right) in Cleft( {{text{R}}_t ,dot H^1 left( {{text{R}}^{text{3}} } right) times L^2 left( {{text{R}}^{text{3}} } right)} right)) with conservation of energy n n